If all of the telephone extensions in a certain company must

Question

If all of the telephone extensions in a certain company must be even numbers, and if each of the extensions uses all four of the digits 1, 2, 3, and 6, what is the greatest number of four-digit extensions that the company can have?

(A) 4
(B) 6
(C) 12
(D) 16
(E) 24

(A) 4
(B) 6
(C) 12
(D) 16
(E) 24

rxs0005 · Accepted Answer

let the 4 digits be

the possible ways are

A B C D

1 * 2 *3 * 2 = 12

total 12 ways

the logic is simple

4 digits are there so the units shud be 2 or 6 tht can be counted in 2 ways

for the next digit there are 3 left so 3 ways the next digit 2 left 2 ways last one can be chosen in 1 way for the lefot over digit

so 1* 2* 3* 2 = 12

P.S. but also if repetetions are allowed then it will be

4* 4 * 4* 2 = 128

GMATPIPO · Answer

nice question,

*** 1st, the number can be expressed as "ABC2" (ending in 2)

so, there are 3*2*1*1 = 6 ways to express the number using 1,2,3,6


*** 2nd, the number can be expressed as "ABC6" (ending in 6)

so, there are 3*2*1*1 = 6 more ways to express the number using 1,2,3,6 


therefore: 6 + 6 = 12

BrentGMATPrepNow · Answer

Take the task of arranging the 4 digits and break it into  stages .

We’ll begin with the  most restrictive stage .

Stage 1 : Select the digit in the units position 
Since the 4-digit extension must be EVEN, the unit digit must be either 2 or 6
So, we can complete stage 1 in  2  ways

Stage 2 : Select the tens digit
There are 3 remaining digits from which to choose, so we can complete this stage in  3  ways.

Stage 3 : Select the hundreds digit
There are 2 digits remaining, so we can complete this stage in  2  ways.

Stage 4 : Select the thousands digit
There 1 digit remaining, so we can complete this stage in  1  way.

By the Fundamental Counting Principle (FCP), we can complete all 4 stages (and thus create an even extension) in  (2)(3)(2)(1)  ways (= 12 ways)

Answer: D

NOTE : the FCP can be used to solve the MAJORITY of counting questions on the GMAT. For more information about the FCP, watch our free video:  https://www.gmatprepnow.com/module/gmat-counting/video/775

Cheers,
Brent

BrentGMATPrepNow · Answer

NOTE : Always  SCAN  the answer choices before performing any calculations. 
Here, we notice that the answer choices are pretty small, so if we don't come up a full solution, we should consider LISTING AND COUNTING

We get the following extensions:
1236
1326
1362
1632
2136
2316
3126
3162
3216 
3612
6132
6312
TOTAL = 12
Answer: C

Cheers,
Brent

JeffTargetTestPrep · Answer

Solution:

If we let 6 be the last digit, then we have 3! = 6 options for the other 3 digits. If we let 2 be the last digit, we have 6 options also. Therefore, we have a total of 12 options.

Answer: C

ScottTargetTestPrep · Answer

If we let 6 be the last digit, then we have 3! = 6 options for the other 3 digits. If we let 2 be the last digit, we have 6 options also. Therefore, we have a total of 12 options.

Answer: C

YinLZ · Answer

Hello,
This is probably a silly question...but why 4 cannot be in the unit place? isn't' 4 also an even number?
Thank you!

Bunuel · Answer

We are told that only 1, 2, 3, and 6 for the extensions, so as you can see 4 is not among the numbers allowed.

Kimberly77 · Answer

Hi  BrentGMATPrepNow , Question asked what is the greatest number of four-digit extensions that the company can have? Which mean numbers doesn't need to be different and digit can repeat except the last digit of either 2 or 6. Therefore not sure why is not 4 * 4 * 4 * 2 = 128 ?

BrentGMATPrepNow · Answer

The question tells us "... each of the extensions uses  all four  of the digits 1, 2, 3, and 6"

Kimberly77 · Answer

Thanks  BrentGMATPrepNow .  all four  here mean it's unique/different. Noted thanks Brent.
In that case, could we presume that when it comes to numbers like telephone extension, codes...etc which position matters. it's presumed to be different?

BrentGMATPrepNow · Answer

I wouldn't say that  all four  implies that the words must be different. 
In general, if a question asks you to determine the total number of outcomes, I would say that it's implied that each outcome is different. 
For example, for this question we wouldn't count the extensions 1263 and 1263 as different outcomes since they are the same numbers.

I would say that in the directions "... each of the extensions uses  all four  of the digits 1, 2, 3, and 6" ,  all four  means: 
The digit 1 must appear in the extension. 
And, the digit 2 must appear in the extension. 
And, the digit 3 must appear in the extension. 
And, the digit 6 must appear in the extension.

Kimberly77 · Answer

Noted. Thanks  BrentGMATPrepNow

EnglishAgast · Answer

Hi  Bunuel  this question appeared in GMAT Focus Prep Test #2. Can you pls add the relevant tag? Thank you!

Bunuel · Answer

______________

Done. Thank you!

saynchalk · Answer

any way to do this using combinations?

If all of the telephone extensions in a certain company must

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